CONSTRAINED HAMILTONIANS AND LOCAL-SQUARE-ROOT ACTIONS

Abstract

The configuration space of general relativity is superspace, the space of Riemannian three-geometries and the Hamiltonian is just a sum of constraints, with Lagrange multipliers. One can go from this Hamiltonian, via a Legandre transformation, back and forth to the Lagrangian. The Lagrange multiplier (the lapse function) can be eliminated from the Lagrangian and one is left with an action which is a product of square roots. This is the Baierlein-Sharp-Wheeler action for general relativity. This action is unique in that all other square root actions are not self-consistent. This paper shows how to express this result in phase space. The only selfconsistent constrained Hamiltonian on superspace which is ultralocal and quadratic in the momenta in the ADM Hamiltonian for general relativity.